Existence and Uniqueness of Path Wise Solutions forStochastic Integral Equations Driven by non Gaussian

Noise on Separable Banach Spaces

V. Mandrekar, B. Rüdiger

no. 186

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-

gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer-

sität Bonn entstanden und als Manuskript vervielfältigt worden.

Bonn, Oktober 2004

Existence and uniqueness of path wise solutionsfor stochastic integral equations driven by nonGaussian noise on separable Banach spaces

V. Mandrekar*, B. Rudiger*** Department of Statistics and Probability Michigan State University, EastLansing, MI 48824, USA** - Mathematisches Institut, Universitat Koblenz-Landau, Campus Koblenz,Universitatsstrasse 1, 56070 Koblenz, Germany;- SFB 611, Institut fur Angewandte Mathematik, Abteilung Stochastik, Univer-sitat Bonn, Wegelerstr. 6, D -53115 Bonn, Germany

Abstract

The stochastic integrals of M- type 2 Banach valued random functionsw.r.t. compensated Poisson random measures introduced in [21] are dis-cussed for general random functions. These are used to solve stochasticintegral equations driven by non Gaussian noise on such spaces. Exis-tence and uniqueness of the path wise solutions are proven under localLipshitz conditions for the drift and noise coefficients on M- type 2 aswell as general separable Banach spaces. The continuous dependence ofthe solution on the initial data as well as on the drift and noise coefficientsare shown. The Markov properties for the solutions are analyzed. Someexample where this theory can be used to solve applied problems, e.g.related to finance, are provided.

AMS -classification (2000): 60H05, 60G51, 60G57, 46B09, 47G99

Keywords: Stochastic differential equations, stochastic integrals on separableBanach spaces, M- type 2 Banach spaces, martingales measures, compensatedPoisson random measures, additive processes, random Banach valued functions

1 Introduction

In this article we shall study existence and uniqueness of the path wise solutionof the following Banach valued integral equation

Zt(ω) = φ(t, ω)+∫ t

0

A(s, Zs(ω), ω)ds+∫ t

0

∫Λ

f(s, x, Zs(ω), ω)(N(dsdx)(ω)−ν(dsdx))

(1)

on each time interval [0, T ], T > 0, where (Zt)t∈IR+ is a random process withvalues in a separable Banach space F . N(dsdx)(ω) − ν(dsdx) is a compen-sated Poisson random measure (cPrm) associated to a Levy process (Xt)t≥0

1

(Definition 2.7 in Section 2). The process (Xt)t≥0 has values in a separa-ble Banach space E and is defined on a probability space (Ω,F , P ), so thatN(dsdx)(ω) (for each ω ∈ Ω fixed) and its compensator ν(dsdx) are σ -finitemeasures on the σ -algebra B(IR+×E \0), generated by the product semiringB(IR+) × B(E \ 0) of the Borel σ -algebra B(IR+) and the trace σ -algebraB(E \ 0). We assume ν(dsdx)= dsβ(dx), where ds denotes the Lesbeguemeasure on B(IR+) (see Section 2 for the precise definitions of the (random)measures N(dsdx)(ω) , ν(dsdx) ). A(t, z, ω) , φ(t, ω) and f(t, x, z, ω) witht ∈ IR+, x ∈ E \ 0, z ∈ F , ω ∈ Ω, are measurable in all their variables andhave values in F (see Section 3 for a precise definition). The stochastic inte-grals

∫ t

0

∫Λg(s, x, ω)(N(dsdx)(ω) − ν(dsdx)), with T > 0, Λ ∈ B(E \ 0) and

g(t, x, ω) with values in a Hilbert and Banach space, have been defined in [21].Here we improve some of these results (see e.g. Theorem 3.13). From theseresults it follows that the integral

∫ t

0

∫Λf(s, x, Zs(ω), ω)(N(dsdx)(ω)−ν(dsdx))

is well defined (see Section 4). We refer to Section 2, where the results of [21],that we need in this article, are reported.In Section 3 we shall prove that if φ(t, ω) does not depend on time, i.e. φ(t, ω) =φ(ω), and satisfies the condition

E[‖φ(ω)‖p] <∞ , (2)

with p = 1, and A(t, z, ω) and f(t, x, z, ω) satisfy a Lipshitz - condition, thereis a unique cad -lag solution of (1) for (t ∈ (0, T ]) s.th.∫ T

0

E[‖Zs‖p] ds <∞ , (3)

with p = 1. (We denote with E[·] the expectation w.r.t. the probability P ).(See Section 3 for a precise statement.) If F is a separable Banach space ofM- type 2 the same results hold for p = 2. We remark that we do not requirethat the integrand f(s, x, z, ω) or the solution Zs has a left continuous versionor is predictable in the sense of [12]. This is a consequence of the fact that wehave defined the stochastic integrals w.r.t. the compensated Poisson randommeasures like in [21], i.e. in a stronger sense than in [12]. (See also Section 3where such integrals are discussed and results of [21] are improved). If φ(t, ω)depends also on time and satisfies the condition∫ T

0

E[‖φ(s, ω)‖p] ds <∞ , (4)

then uniqueness of a cad -lag solution (1) for t ∈ (0, T ] is also proven, but onlyup to stochastic equivalence.We recall here the definition of M -type 2 separable Banach space (see e.g. [18]).

2

Definition 1.1 A separable Banach space F , with norm ‖ · ‖, is of M -type 2,if there is a constant K2, such that for any F - valued martingale (Mk)k∈1,..,n

the following inequality holds:

E[‖Mn‖2] ≤ K2

n∑k=1

E[‖Mk −Mk−1‖2] , (5)

with the convention that M−1 = 0.

Definition 1.2 A separable Banach space F is of type 2, if there is a constantK2, such that if Xin

i=1 is any finite set of centered independent F - valuedrandom variables, such that E[‖Xi‖2] <∞, then

E[‖n∑

i=1

Xi‖2] ≤ K2

n∑i=1

E[‖Xi‖2] (6)

We remark that any separable Banach space of M -type 2 is a separable Banachspace of type 2. Moreover, a separable Banach space is of type 2 as well as ofcotype 2 if and only if it is isomorphic to a separable Hilbert space [16], wherea Banach space of cotype 2 is defined by putting ≥ instead of ≤ in (6) (see [3],or [17]).

2 Poisson and Levy measures of additive pro-cesses on separable Banach spaces

We assume that a filtered probability space (Ω,F , (Ft)0≤t≤+∞, P ), satisfyingthe ”usual hypothesis”, is given:i) Ft contain all null sets of F , for all t such that 0 ≤ t < +∞ii) Ft = F+

t , where F+t = ∩u>tFu, for all t such that 0 ≤ t < +∞, i.e. the

filtration is right continuous

In this Section we introduce the compensated Poisson random measures asso-ciated to additive processes on (Ω,F , (Ft)0≤t≤+∞, P ) with values in (E,B(E)),where in the whole paper we assume that E is a separable Banach space withnorm ‖ · ‖ and B(E) is the corresponding Borel σ -algebra.

Definition 2.1 A process (Xt)t≥0 with state space (E,B(E)) is an Ft - additiveprocess on (Ω,F , P ) ifi) (Xt)t≥0 is adapted (to (Ft)t≥0)ii) X0 = 0 a.s.iii) (Xt)t≥0 has increments independent of the past, i.e. Xt−Xs is indepen-dent of Fs if 0 ≤ s < tiv) (Xt)t≥0 is stochastically continuous, i.e. ∀ε > 0 lims→t P (‖Xs −Xt‖ >ε) = 0.

3

v) (Xt)t≥0 is cadlag.

An additive process is a Levy process if the following condition is satisfiedvi) (Xt)t≥0 has stationary increments, that is Xt −Xs has the same distri-bution as Xt−s, 0 ≤ s < t.

Let (Xt)t≥0 be an additive process on (E,B(E)) (in the sense of Definition 2.1).Set Xt− := lims↑tXs and ∆Xs := Xs −Xs− .

The following results, i.e. Theorem 2.2, Theorem 2.3, Corollary 2.4, Theorem2.5, Corollary 2.6 are known (see e.g. [10]). (The proofs of Theorem 2.3, Corol-lary 2.4, Theorem 2.5, Corollary 2.6 can be given following [2]).

Theorem 2.2 Let Λ ∈ B(E), 0 ∈ (Λ)c (where as usual Γ denotes the closureof the set Γ and with Γc we denote the complementary of a set Γ), then

NΛt :=

∑0<s≤t

1Λ(∆Xs) =∑n≥1

1t≥TΛn

(7)

where

TΛ1 := infs > 0 : ∆Xs ∈ Λ (8)

TΛn+1 := infs > Tn

Λ : ∆Xs ∈ Λ, n ∈ IN. (9)

NΛt is an adapted counting process without explosions and

P (NΛt = k) = exp(−νt(Λ))

(νt(Λ))k

k!(10)

νt(Λ) := E[NΛt ] (11)

Theorem 2.3 Let B(E \ 0) be the trace σ -algebra on E \ 0 of the Borel σ-algebra B(E) on E, and let

F(E \ 0) := Λ ∈ B(E \ 0) : 0 ∈ (Λ)c , (12)

then F(E \ 0) is a ring and for all ω ∈ Ω the set function

N ·t := Nt(ω, ·) : F(E \ 0) → IR+ (13)

Λ → NΛt (ω)

is a σ -finite pre- measure (in the sense of e.g. [4]).

4

Corollary 2.4 For any ω ∈ Ω there is a unique σ -finite measure on B(E \0)

Nt(ω, ·) : B(E \ 0) → IR+ (14)

Λ → NΛt (ω)

which is the continuation of the σ -finite pre- measure on F(E \ 0) given byTheorem 2.3.

From Theorem 2.3, Corollary 2.4 it follows that Nt : Λ → NΛt is a random

measure on (E \ 0,B(E \ 0)).

Theorem 2.5 The set function νt(Λ) := E[NΛt (ω)] ∈ IR, Λ ∈ F(E \ 0),

ω ∈ Ω satisfies:

νt : F(E \ 0) → IR+ (15)

Λ → E[NΛt (ω)]

and is a σ -finite pre -measure on ((E \ 0),F(E \ 0))

Corollary 2.6 There is a unique σ -finite measure on the σ -algebra B(E \0)

νt : B(E \ 0) → IR+ (16)

Λ → E[NΛt (ω)]

which is the continuation to B(E \ 0) of the σ -finite pre- measure νt on thering ((E \ 0),F(E \ 0)), given by Theorem 2.5.

Let S(IR+) be the semi -ring of sets (t1, t2], 0 ≤ t1 < t2, and S(IR+)×B(E\0)be the semi -ring of the product sets (t1, t2]× Λ, Λ ∈ B(E \ 0).Let

N((t1, t2]× Λ)(ω) = Nt2(Λ)(ω)−Nt1(Λ)(ω) ∀Λ ∈ B(E \ 0) ∀ω ∈ Ω (17)

For all ω ∈ Ω fixed, N(dtdx)(ω) is a σ -finite pre -measure on the product semi-ring S(IR+)× B(E \ 0).Let us denote also by N(dtdx)(ω) the measure which is the unique extension ofthe pre -measure to the σ -algebra B(IR+ × (E \ 0)) generated by S(IR+) ×B(E \ 0) (see e.g. [4] Satz 5.7, Chapt. I, §5, [15] Theorem 1, Chapt. V, §2 forthe existence of a unique minimal σ -algebra containing a product semi -ring).

Let

ν((t1, t2]× Λ) = νt2(Λ)− νt1(Λ) ∀A ∈ B(E \ 0) (18)

ν(dtdx) is a σ -finite pre -measure on S(IR+)×B(E \0). Let us denote also byν(dtdx) the σ -finite measure, which is the unique extension of this pre -measureon B(IR+ × (E \ 0)).

5

Definition 2.7 We call N(dtdx)(ω) the Poisson random measure associated tothe additive process (Xt)t≥0 and ν(dtdx) its compensator. We call N(dtdx)(ω)−ν(dtdx) the compensated Poisson random measure associated to the additiveprocess (Xt)t≥0. (We omit sometimes to write the dependence on ω ∈ Ω.)

Remark 2.8 Let N(dtdx)(ω) − ν(dtdx) be the compensated Poisson randommeasure associated to an additive process (Xt)t≥0 with values in a Banach spaceE defined on the measure space (E \ 0,B(E \ 0). (Xt)t≥0 is a Levy processiff ν(dtdx)= dtβ(dx), where dt denotes the Lesbegues measure on B(IR+), andβ(dx) is a σ -finite measure on (E \0,B(E \0)), and is called Levy measureassociated to (Xt)t≥0.

3 Stochastic integrals w.r.t. compensated Pois-son random measures

Let N(dtdx)(ω) − ν(dtdx) be the compensated Poisson random measure asso-ciated to an additive process (Xt)t≥0 defined on a (Ω,F , (Ft)t≥0, P ) and withvalues in a separable Banach space E (Definition 2.7 in Section 2).

Let F be a separable Banach space with norm ‖·‖F . (When no misunderstandingis possible we write ‖ · ‖ instead of ‖ · ‖F .) Let Ft := B(IR+ × (E \ 0)) ⊗ Ft

be the product σ -algebra generated by the semi -ring B(IR+ × (E \ 0))×Ft

of the product sets Λ× F , Λ ∈ B(IR+ × E \ 0), F ∈ Ft. Let T > 0, and

MT (E/F ) :=f : IR+ × E \ 0 × Ω → F, such that f is FT /B(F ) measurable

f(t, x, ω) is Ft − adapted ∀x ∈ E \ 0, t ∈ (0, T ] (19)

In this Section we shall introduce the stochastic integrals of random functionsf(t, x, ω) ∈ MT (E/F ) with respect to the compensated Poisson random mea-sures q(dtdx)(ω) := N(dtdx)(ω) − ν(dtdx) associated to an additive process(Xt)t≥0 discussed in [21]. (We omit sometimes to write the dependence onω ∈ Ω.)

There is a ”natural definition” of stochastic integral w.r.t. q(dtdx)(ω) on thosesets (0, T ] × Λ where the measures N(dtdx)(ω) (with ω fixed) and ν(dtdx) arefinite, i.e. 0 /∈ Λ. According to [21] (see also [2] for the case of deterministicfunctions f(x) , x ∈ \0 ) we give the following definition

6

Definition 3.1 Let t ∈ (0, T ], Λ ∈ F(E\0) (defined in (12)), f ∈MT (E/F ).Assume that f(·, ·, ω) is Bochner integrable on (0, T ]×Λ w.r.t. ν, for all ω ∈ Ωfixed. The natural integral of f on (0, t] × Λ w.r.t. the compensated Poissonrandom measure q(dtdx) := N(dtdx)(ω)− ν(dtdx) is∫ t

0

∫Λf(s, x, ω) (N(dsdx)(ω)− ν(dsdx)) :=∑

0<s≤t f(s, (∆Xs)(ω), ω)1Λ(∆Xs(ω))−∫ t

0

∫Λf(s, x, ω)ν(dsdx) ω ∈ Ω(20)

where the last term is understood as a Bochner integral, (for ω ∈ Ω fixed) off(s, x, ω) w.r.t. the measure ν.

It is more difficult to define the stochastic integral on those sets (0, T ]×Λ, Λ ∈B(E \ 0), s.th. ν((0, T ]×Λ) = ∞. For real valued functions this problem wasalready discussed e.g. in [24] and [6], [12], [25] (for general martingale measures).Different definitions of stochastic integrals were proposed. In [21] (and [2] for thecase of deterministic functions f(x) ,x ∈ E \ 0 ) we introduced the definitionof ”strong -p - integral” (Definition 3.8). The strong -p -integral is the limit inLF

p (Ω,F , P ) (the space of F -valued random variables Y , with E[‖Y ‖p] < ∞,defined in Definition 3.7) of the ”natural integrals” (20) of the ”simple functions”defined in (21). (We refer to Definition 3.8 for a precise statement.) If p = 2, thisconcept generalizes the definition in [6] of stochastic integration of real valuedfunctions with respect to martingales measures on IRd, to the case of Banachspace valued functions, for the case where the martingale measures are givenby compensated Poisson random measures on general separable Banach spaces.It generalizes also to the stochastic integral introduced in [24] for real valuedfunctions integrated w.r.t. compensated Poisson random measures associated toα -stable Levy processes on IR. In [21] it has also been proven that the conceptof strong -p -integral, with p = 1 or p = 2, is more general than the definition ofstochastic integrals w.r.t. point processes introduced (for the real valued case)in [12], Chapt. II.3. In fact, for the existence of the strong -p -integral, withp = 1 or p = 2, no predictability condition (in the sense of [12]) is needed forthe integrand. This condition is however needed for the stochastic integralsintroduced in [12], which in [21] are denoted with ”simple -p -integrals”. (Thisconcept has been generalized to the Hilbert or Banach valued case in [21], too.)If the integrand is however left -continuous, then the strong -p-integral coincidewith the simple -p- integral (we refer to [21] for a precise statement).We recall here the definition of ”strong -p -integral”, p ≥ 1, introduced in [21].We first introduce the simple functions.

Definition 3.2 A function f belongs to the set Σ(E/F ) of simple functions ,if f ∈MT (E/F ), T > 0 and there exist n ∈ IN , m ∈ IN , such that

f(t, x, ω) =n−1∑k=1

m∑l=1

1Ak,l(x)1Fk,l

(ω)1(tk,tk+1](t)ak,l (21)

7

where Ak,l ∈ F(E \ 0) (i.e. 0 /∈ Ak,l), tk ∈ (0, T ], tk < tk+1, Fk,l ∈ Ftk,

ak,l ∈ F . For all k ∈ 1, ..., n− 1 fixed, Ak,l1 × Fk,l1 ∩Ak,l2 × Fk,l2= ∅ if l1 6= l2.

Proposition 3.3 Let f ∈ Σ(E/F ) be of the form (21), then∫ T

0

∫A

f(t, x, ω)q(dtdx)(ω) =n−1∑k=1

m∑l=1

ak,l1Fk,l(ω)q((tk, tk+1]∩(0, T ]×Ak,l ∩A)(ω).

(22)for all A ∈ B(E \ 0), T > 0.

Remark 3.4 The random variables 1Fk,lin (22) are independent of q((tk, tk+1]∩

(0, T ]×Ak,l ∩A) for all k ∈ 1...n− 1, l ∈ 1...m fixed.

Proof of Proposition 3.3: The proof is an easy consequence of the Definition2.7 of the random measure q(dtdx)(ω).

We recall here the definition of strong -p -integral, p ≥ 1, (Definition 3.8 below)given in [21] through approximation of the natural integrals of simple functions.First we establish some properties of the functions f ∈MT,p

ν (E/F ), where

MT,pν (E/F ) := f ∈MT (E/F ) :

∫ T

0

∫E[‖f(t, x, ω)‖p] ν(dtdx) <∞ (23)

Theorem 3.5 [21] Let p ≥ 1. Suppose that the compensator ν(dtdx) of thePoisson random measure N(dtdx) satisfies the following hypothesis A.

Hypothesis A: ν is a product measure ν = α⊗ β on the σ -algebra generated bythe semi -ring S(IR+) × B(E \ 0), of a σ -finite measure α on S(IR+), s.th.α([0, T ]) <∞ , ∀T > 0 , α is absolutely continuous w.r.t the Lesbegues measureon IR+, and a σ -finite measure β on B(E \ 0).

Let T > 0, then for all f ∈ MT,pν (E/F ) and all Λ ∈ B(E \ 0), there is a

sequence of simple functions fnn∈IN satisfying the following property :

Property P: fn ∈ Σ(E/F )∀n ∈ IN , fn converges ν ⊗ P -a.s. to f on (0, T ] ×Λ× Ω, when n→∞, and

limn→∞

∫ T

0

∫Λ

E[‖fn(t, x)− f(t, x)‖p] dν = 0 , (24)

i.e. ‖fn − f‖ converges to zero in Lp((0, T ]× Λ× Ω, ν ⊗ P ), when n→∞.

Definition 3.6 We say that a a sequence of functions fn are Lp -approximatingf if these satisfy property P, i.e. fn converge ν⊗P -a.s. to f on (0, T ]×Λ×Ω,when n→∞, and satisfy (24).

8

For the real valued case Theorem 3.5 has also been stated in in Chapt.2 Section4 [24] or in [6], without proof.

Definition 3.7 Let p ≥ 1, LFp (Ω,F , P ) is the space of F -valued random vari-

ables, such that E‖Y ‖p =∫‖Y ‖pdP < ∞. We denote by ‖ · ‖p the norm

given by ‖Y ‖p = (E‖Y ‖p)1/p. Given (Yn)n∈IN , Y ∈ LFp (Ω,F , P ), we write

limpn→∞ Yn = Y if limn→∞ ‖Yn − Y ‖p = 0.

In [21] we introduce the following

Definition 3.8 Let p ≥ 1, t > 0. We say that f is strong -p -integrable on(0, t] × Λ, Λ ∈ B(E \ 0), if there is a sequence fnn∈IN ∈ Σ(E/F ), whichsatisfies the property P in Theorem 3.5, and such that the limit of the naturalintegrals of fn w.r.t. q(dtdx) exists in LF

p (Ω,F , P ) for n→∞, i.e.∫ t

0

∫Λ

f(t, x, ω)q(dtdx)(ω) :=p

limn→∞

∫ t

0

∫Λ

fn(t, x, ω)q(dtdx)(ω) (25)

exists. Moreover, the limit (25) does not depend on the sequence fnn∈IN ∈Σ(E/F ), for which property P and (25) holds.We call the limit in (25) the strong -p -integral of f w.r.t. q(dtdx) on (0, t]×Λ.

Remark 3.9 In [21] it has been proven that the strong -p -integrals are mar-tingales. It could then be concluded that these have a cad -lag version (see e.g.[9], [13],[20]). We shall propose in Proposition 3.15 another proof of this state-ment, for the case of the strong -p -integrals of functions f(t, x, ω), which arein MT,p

ν (E/F ), with p = 1, or p = 2 in case where F is a Banach space of M-type 2. The proof of 3.15 includes for these cases also the proof of the strongerstatement that under the above conditions the strong -p- integrals are cad -lag.

We now give sufficient conditions for the existence of the strong -p -integrals,when p = 1, or p = 2. In the whole article we assume that hypothesis A inTheorem 3.5 is satisfied.

Theorem 3.10 [21] Let f ∈MT,1ν (E/F ), then f is strong -1 -integrable w.r.t.

q(dt, dx) on (0, t]× Λ, for any 0 < t ≤ T , Λ ∈ B(E \ 0) . Moreover

E[‖∫ t

0

∫Λ

f(s, x, ω)q(dsdx)(ω)‖] ≤ 2∫ t

0

∫Λ

E[‖f(s, x, ω)‖]ν(dsdx)(ω) (26)

Remark 3.11 By definition of Bochner integral f ∈MT,1ν (E/F ) iff f ∈MT (E/F )

and f is Bochner integrable w.r.t. ν ⊗ P . Moreover, from Definition 3.8 andTheorem 3.10 it also follows that f is strong -1 -integrable, iff f ∈ MT (E/F )and f is Bochner integrable w.r.t. ν ⊗ P .

9

Theorem 3.12 [21] Suppose (F,B(F )):= (H,B(H)) is a separable Hilbert space.Let f ∈MT,2

ν (E/H), then f is strong 2 -integrable w.r.t. q(dtdx) on (0, t]× Λ,for any 0 < t ≤ T , Λ ∈ B(E \ 0). Moreover

E[‖∫ t

0

∫Λ

f(s, x, ω)q(dsdx)(ω)‖2] =∫ t

0

∫Λ

E[‖f(s, x, ω)‖2]ν(dsdx) (27)

The following Theorem 3.13 was proven in [21] for the case of deterministicfunctions on type 2 Banach spaces, and on M-type 2 spaces for functions whichdo not depend on the random variable x. We extend it here to general randomfunctions on any M- type 2 Banach space.

Theorem 3.13 Suppose that F is a separable Banach space of M- type 2. Letf ∈MT,2

ν (E/F ), then f is strong 2 -integrable w.r.t. q(dtdx) on (0, t]× Λ, forany 0 < t ≤ T , Λ ∈ B(E \ 0). Moreover

E[‖∫ t

0

∫Λ

f(s, x, ω)q(dsdx)(ω)‖2] ≤ K22

∫ t

0

∫Λ

E[‖f(s, x, ω)‖2]ν(dsdx) . (28)

where K2 is the constant in the Definition 1.1 of M -type 2 Banach spaces.

Proof of Theorem 3.13:

First we prove that given f ∈ Σ(E/F ) the inequality (28) holds:let f be of the form (21), it then follows from Proposition 3.3

E[‖∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω)‖2]

= E[‖∑n−1

k=1

∑nl=1 1Fk,l

ak,lq((tk, tk+1] ∩ (0, t]×Ak,l ∩ Λ)‖2]

≤ K2E[∑n−1

k=1 ‖∑n

l=1 1Fk,lak,lq((tk, tk+1] ∩ (0, t]×Ak,l ∩ Λ)‖2

](29)

where in the inequality we used the M- type 2 condition and the fact that∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω) is a martingale [21].

In order to bound the r.h.s. of (29) by putting the norm inside all the sumoperators, we decompose for each k ∈ 1, ..., n − 1 fixed the sets ∪m

l=1Ak,l indisjoint sets, so that we can then use once more the M -type 2 condition andprove the inequality (28) as described below:let 21,...,m be the set of all subsets of the set 1, ...,m, then

∪ml=1Ak,l = ∪j1,...,jl∈21,...,mAj1 ∩ ... ∩Ajl

\ ∪mh=1,h 6=j1,...jl

Ah , (30)

and the sets Aj1∩...∩Ajl\∪m

h=1,h 6=j1,...jlAh , are two by two disjoint. To facilitate

the notations we define

10

qk,A(ω) := q((tk, tk+1] ∩ (0, T ]×A ∩ Λ)(ω) .

We then have

E[‖∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω)‖2]

≤ K2E[∑n−1

k=1 ‖∑n

l=1 1Fk,lak,lqk,Ak,l

(ω)‖2]

≤ K2E[∑n−1

k=1 ‖∑

j1,...,jl∈21,...,m

(ak,j11Fk,j1

(ω) + ...+ ak,jl1Fk,jl

(ω))×

qk,Ak,j1∩...∩Ak,jl\∪h=1,h 6=j1,...,jl

Ak,h(ω)‖2]

= K2E[∑n−1

k=1 EFtk[‖∑

j1,...,jl∈21,...,m

(ak,j11Fk,j1

(ω) + ...+ ak,jl1Fk,jl

(ω))×

qk,Ak,j1∩...∩Ak,jl\∪h=1,h6=j1,...,jl

Ak,h(ω)‖2]]

≤ K22E[

∑n−1k=1 EFtk

[∑

j1,...,jl∈21,...,m ‖ak,j11Fk,j1(ω) + ...+ ak,jl

1Fk,jl(ω)‖2 ×

(qk,Ak,j1∩...∩Ak,jl\∪h=1,h 6=j1,...,jl

Ak,h(ω))2]]

≤ K22E[

∑n−1k=1 EFtk

[∑

j1,...,jl∈21,...,m (‖ak,j1‖1Fk,j1(ω) + ...+ ‖ak,jl

‖1Fk,jl(ω)‖)2 ×

(qk,Ak,j1∩...∩Ak,jl\∪h=1,h 6=j1,...,jl

Ak,h(ω))2]]

= K22E[

∑n−1k=1 EFtk

[∑

j1,...,jl∈21,...,m

(‖ak,j1‖21Fk,j1

(ω) + ...+ ‖ak,jl‖21Fk,jl

(ω))×

(qk,Ak,j1∩...∩Ak,jl\∪h=1,h6=j1,...,jl

Ak,h(ω))2]]

= K22E[

∑n−1k=1 EFtk

[∑m

l=1 1Fk,l(ω)‖ak,l‖2|qk,Ak,l

(ω)|2]]

= K22

∑n−1k=1

∑ml=1 P (Fk,l)‖ak,l‖2ν((tk, tk+1] ∩ (0, t]× Λ ∩Ak,l)

=∫ t

0

∫ΛE[‖f(s, x‖2]ν(dsdx) (31)

In the above calculations we used the M -type 2 condition applied to the inde-pendent random variables qk,Ak,j1∩...∩Ak,jl

\∪h=1,h 6=j1,...,jlAk,h

(ω). (In fact for thispassage the type 2 condition would be sufficient.) Moreover we used that

(‖ak,j1‖1Fk,j1(ω) + ...+ ‖ak,jl

‖1Fk,jl(ω)‖)2(qk,Ak,j1∩...∩Ak,jl

\∪h=1,h6=j1,...,jlAk,h

(ω))2 =

(‖ak,j1‖21Fk,j1(ω) + ...+ ‖ak,jl

‖21Fk,jl(ω))(qk,Ak,j1∩...∩Ak,jl

\∪h=1,h6=j1,...,jlAk,h

(ω))2(32)

which follows from

‖ak,jp‖1Fk,jp

(ω)‖ak,ji‖1Fk,ji

(ω)(qk,Ak,j1∩...∩Ak,jl\∪h=1,h 6=j1,...,jl

Ak,h(ω))2 = 0 ∀ji, jp ∈ j1, ..., jl, i 6= p

and is a consequence of

Fk,i ×Ak,i ∩ Fk,p ×Ak,p = ∅ ∀i, p ∈ 1, ...,m .

11

Now we prove that inequality (28) holds for any f ∈MT,2ν (E/F ):

let fnn∈IN ∈ Σ(E/F ) be a sequence L2 -approximating f in (0, T ] × A × Ωw.r.t. ν ⊗ P . Then

E[‖∫ t

0

∫A

(fn(s, ω)−fm(s, ω))q(dsdx)(ω)‖2] ≤ K22

∫A

E[‖fn(s, ω)−fm(s, ω)‖2]ν(dsdx)

(33)

so that∫ t

0

∫Afn(s, ω)q(dsdx)(ω) is a Cauchy sequence in LF

2 (Ω,F , P ) and thelimit (25) for p = 2 exists. Moreover the limit does not depend on the choice ofthe sequence fnn∈IN . It follows that the strong - 2- integral

∫ t

0

∫Af(s, x, ω)q(dsdx)(ω)

exists. Moreover

E[‖∫ t

0

∫Af(s, x, ω)q(dsdx)(ω)‖2] = limn→∞E[‖

∫ t

0

∫Afn(s, x, ω)q(dsdx)(ω)‖2]

≤ limn→∞K22

∫ t

0

∫AE[‖fn(s, x, ω)‖2]ν(dsdx) = K2

2

∫ t

0

∫AE[‖f(s, x, ω)‖2]ν(dsdx)

Remark 3.14 Let 0 /∈ Λ. Suppose that the hypothesis of Theorem 3.10, or ofTheorem 3.12, or of Theorem 3.13 are satisfied. Suppose that f(·, x, ω) is left-continuous for all x ∈ E, and P -a.e. ω ∈ Ω. From Corollary 5.2 in [21] itfollows that the strong -p- integral (with p = 1 in case of Theorem 3.10, andp = 2, in case of the Theorems 3.12 -3.13) coincides P -a.s. with the naturalintegral. If the condition that f(·, x, ω) is left -continuous for all x ∈ E and P-a.e. ω ∈ Ω is not satisfied, then this might be false (see the Proof of Theorem5.1 in [21]).

Proposition 3.15 Let f satisfy the hypothesis of Theorem 3.10, or 3.13. Then∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω) , t ∈ [0, T ] is an Ft -martingale with mean zero and

is cad -lag.

Proof of Proposition 3.15: Let p = 1 if the hypothesis of Theorem 3.10 aresatisfied, p = 2 if the hypothesis of Theorem 3.13 are satisfied. From Proposition3.3 it follows that the natural integral

∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω) of a simple

function f(s, x, ω) is P -a.s. cad -lag in t and is an Ft -martingale. It followsthat ‖

∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω)‖ is a submartingale. Using Doob’ s inequality

we get for p = 1, resp. p = 2,

P (sup0≤t≤T ‖∫ t

0

∫Λf(s, x, ω)q(dsdx)(ω)‖ > ε) ≤ 1

εpE[‖∫ T

0

∫Λf(s, x, ω) q(dsdx)(ω)‖p]

≤ C 1εp

∫ T

0

∫ΛE[‖f(s, x, ω)‖p] , (34)

12

where the constant C in the last inequality equals C = 2, in case of p = 1, andC = K2, in case of p = 2. The last inequality follows in fact from inequality(26), resp. (28).By linearity of the integral we get the above inequality for fn − fm, n,m ∈ IN ,where ‖fn − f‖ converges to zero in Lp([0, T ] × Λ × Ω, ν ⊗ P ). Hence we getthat

∫ t

0

∫Λf(s, x, ω) q(dsdx)(ω) is a martingale and is cad - lag a.e..

4 Existence and uniqueness of stochastic inte-gral equations driven by non Gaussian noise

In this Section we use the same notations as in the previous Sections. We assumein the whole paper, that the compensator ν(dtdx) of the Poisson random mea-sure N(dtdx)(ω) is a product measure on B(IR+×E), i.e. ν(dtdx) = α(dt)β(dx),such that hypothesis A in Theorem 3.5 is satisfied. Moreover, we assume start-ing from this Section that α(dt) = dt, i.e. N(dtdx)(ω) is a Poisson randommeasure associated to an E -valued Levy process.

We also denote by B([0, T ]×F ) the product σ -algebra generated by the productsemiring B([0, T ]) × B(F ) of the Borel σ -algebra B([0, T ]) and the Borel σ -algebra B(F ) of F , by B([0, T ]×E \ 0×F ) the product σ -algebra generatedby the product semiring B([0, T ] × E \ 0) × B(F ). Given in general two σ-algebras M and L, with measure m and resp. l, we denote by M⊗ L theproduct σ -algebra generated by the product semiring M×L, and by m⊗ l thecorresponding product measure.

We make the following hypothesis:

A) f(t, x, z, ω) is a B([0, T ]×E \ 0 × F )⊗FT /B(F ) -measurable function,s.th. for all t ∈ [0, T ], x ∈ E and z ∈ F fixed, f(t, x, z, ·) is Ft -adapted,B) A(t, z, ω) is a B([0, T ]× F )⊗FT /B(F ) -measurable function, s.th. for allt ∈ [0, T ] and z ∈ F fixed, A(t, z, ·) is Ft -adapted,C) φ(t, ω)∈MT (E/F ) (and does not depend on the variable x ∈ E).

We shall give in this Section sufficient conditions for the existence of a uniquesolution of the integral equation (1), which is in the space LT

p , with p = 1 orp = 2, where

LTp := LT

p ([0, T ]× Ω , (Ft)t∈[0,T ]) :=(Zt(ω))t∈[0,T ], ω ∈ Ω : ∀t ∈ [0, T ] , Zt(ω) is Ft − adapted

and as a function from [0, T ]× Ω to F , is B([0, T ])⊗FT /B(F ) −measurable ,

(Zt(ω))t∈[0,T ] satisfies (3)

13

Definition 4.1 We say that two processes Z1t (ω)∈ LT

p and Z2t (ω) ∈ LT

p aredt⊗ P - equivalent if these coincide for all (t, ω) ∈ Γ, with Γ ∈ B([0, T ])⊗ FT ,and dt⊗P (Γc) = 0, where with dt we denote the Lesbegues measure on B(IR+).We denote with LT

p the set of dt⊗ P -equivalence classes.

Remark 4.2 LTp , p ≥ 1, with norm

‖Zt‖F,T := (∫ T

0

E[‖Zs‖p]ds)1/p . (35)

is a Banach space.

Theorem 4.3 Let 0 < T < ∞ and Λ ∈ B(E \ 0). Let (4) be satisfied forp = 1. Suppose that there is a constant L > 0, s.th.

‖A(t, z, ω)−A(t, z′, ω)‖+∫Λ‖f(t, x, z, ω)− f(t, x, z′, ω)‖β(dx) ≤ L‖z − z′‖

for all t ∈ (0, T ] , z, z′ ∈ F , and for P − a.e. ω ∈ Ω (36)

Assume also that there is a constant K > 0 such that

‖A(t, z, ω)‖+∫Λ‖f(t, x, z, ω)‖β(dx) ≤ K(‖z‖+ 1)

for all t ∈ (0, T ] , z ∈ F and P − a.e. ω ∈ Ω. (37)

then there is a unique process (Zt)0≤t≤T ∈ LT1 which satisfies (1).

Corollary 4.4 Suppose that all hypothesis of Theorem 4.3 are satisfied. Thenthere is up to stochastic equivalence (see Definition 4.5 below) a unique process(Zt)0≤t≤T ∈ LT

1 which satisfies (1). Assume also that φ(t, ω) has P -a.s. nodiscontinuities of the second kind (resp. is cad -lag), then (Zt)0≤t≤T has nodiscontinuities of the second kind (resp. is cad -lag).

The following Definition is well known.

Definition 4.5 Two processes (Xt)t∈IR+ and (Yt)t∈IR+ are stochastic equivalentif P (Xt = Yt) = 1 ∀t ∈ IR+.

Theorem 4.6 Suppose that F is a separable Banach space of M -type 2. Let0 < T < ∞ and Λ ∈ B(E \ 0). Let (4) be satisfied for p = 2. Suppose alsothat there is a constant L > 0, s.th.

14

T‖A(t, z, ω)−A(t, z′, ω)‖2 +∫Λ‖f(t, x, z, ω)− f(t, x, z′, ω)‖2β(dx) ≤ L‖z − z′‖2

for all t ∈ (0, T ] , z, z′ ∈ F , and for P − a.e. ω ∈ Ω (38)

Assume also that there is a constant K > 0 such that

‖A(t, z, ω)‖2 +∫Λ‖f(t, x, z, ω)‖2β(dx) ≤ K(‖z‖2 + 1)

for all t ∈ (0, T ] , z ∈ F and P − a.e. ω ∈ Ω. (39)

then there is a unique process (Zt)0≤t≤T ∈ LT2 which satisfies (1).

Corollary 4.7 Suppose that all hypothesis of Theorem 4.6 are satisfied. Thenthere is up to stochastic equivalence a unique process (Zt)0≤t≤T ∈ LT

2 whichsatisfies (1). If φ(t, ω) has P -a.s. no discontinuities of the second kind (resp.is cad -lag), then (Zt)0≤t≤T has no discontinuities of the second kind (resp. iscad -lag).

(To prove Theorem 4.3 and 4.6 we follow the strategy proposed in [24] for thecase where Zt is real valued and N(dtdx)(ω)−ν(dtdx) is a compensated Poissonrandom measure associated to an α -stable Levy process.)

Proof of Theorem 4.3: Let us consider the mapping S

SZt(ω) := φ(t, ω) +∫ t

0

A(s, Zs(ω), ω)ds+∫ t

0

∫Λ

f(s, x, Zs(ω), ω)q(dsdx)(ω)

(40)

We first prove that, if (Zt(ω))t∈[0,T ]∈ LT1 , then (SZt(ω))t∈[0,T ] ∈ LT

1 .

First let us prove that the integrals in the r.h.s. of (40) are well defined.

That A(s, Zs(ω), ω) is Bochner integrable w.r.t. the Lesbegues measure ds on[0, T ], for P -a.e. ω ∈ Ω, follows from∫ T

0

‖A(s, Zs(ω), ω)‖ds ≤ K

(T +

∫ T

0

‖Zs(ω)‖ds

), (41)

which is a consequence of (37).

That f(s, x, Zs(ω), ω) is strong -1-integrable w.r.t. q(dsdx)(ω) on (0, T ] × Λfollows from (37), Theorem 3.10, and the following inequality∫ T

0

∫Λ

E[‖f(s, x, Zs)‖]ν(dsdx) ≤ K

(T +

∫ T

0

E[‖Zs(ω)‖]ds

), (42)

15

From (41) and (42) it also follows that (SZt(ω))t∈[0,T ] ∈ LT1 . In fact,

∫ T

0dtE[‖SZt‖] ≤

∫ T

0dtE[‖Φ(t, ω)‖] +

∫ T

0dtE[

∫ t

0ds‖A(s, Zs(ω), ω)‖]

+∫ T

0dtE[‖

∫ t

0

∫Λf(s, Zs(ω), x, ω)q(dsdx)(ω)‖]

≤∫ T

0dtE[‖Φ(t, ω)‖] +K(T 2 +

∫ T

0dt∫ t

0dsE[‖Zs(ω)‖]) + 2

∫ T

0dt∫ t

0

∫AE[‖f(s, x, Zs)‖]ν(dsdx)

≤∫ T

0dtE[Φ(t, ω)] + 3K(T 2 +

∫ T

0dt∫ t

0dsE[‖Zs‖]) ≤ 3K(T 2 + T

∫ T

0dsE[‖Zs‖]) <∞ , (43)

where we used again Theorem 3.10.

We shall prove that the operator Sn is a contraction operator on LT1 , for suffi-

ciently large values of n ∈ IN .

Given two processes (Z1t )0≤t≤T , (Z2

t )0≤t≤T ∈ LT1 we have

SZ1t (ω)− SZ2

t (ω) = (44)∫ t

0[A(s, Z1

s (ω), ω)−A(s, Z2s (ω), ω)] ds+

∫ t

0

∫Λ[f(s, x, Z1

s (ω), ω)− f(s, x, Z2s (ω), ω)] q(dsdx)(ω)

From (36) and the properties of Bochner integrals (see e.g. [28] , Chapt. V, §5for such properties), resp. Theorem 3.10, it follows

‖∫ t

0(A(s, Z1

s (ω), ω)−A(s, Z2s (ω), ω)) ds‖

≤∫ t

0‖A(s, Z1

s (ω), ω)−A(s, Z2s (ω), ω)‖ ds

≤ L∫ t

0‖Z1

s (ω)− Z2s (ω)‖ds (45)

E[‖∫ t

0

∫Λf(s, x, Z1

s )− f(s, x, Z2s ) q(dsdx)(ω)‖]

≤ 2∫ t

0

∫ΛE[‖f(s, x, Z1

s )− f(s, x, Z2s )‖] ν(dsdx)

≤ 2L∫ t

0E[‖Z1

s − Z2s‖]ds <∞ (46)

From (44), (45), (46) it follows

E[ ‖SZ1t (ω)− SZ2

t (ω)‖ ]

≤∫ t

0E[‖A(s, Z1

s (ω), ω)−A(s, Z2s (ω), ω)‖] ds

+2∫ t

0

∫ΛE[‖f(s, x, Z1

s (ω), ω)− f(s, x, Z2s (ω), ω)‖] ν(dsdx)

≤ 3L∫ t

0E[‖Z1

s − Z2s‖]ds (47)

It follows by induction that

16

∫ T

0E[ ‖SnZ1

t (ω)− SnZ2t (ω)‖ ]dt (48)

≤ 3nLn∫ T

0dt∫ t

0ds1

∫ s1

0ds2....E[‖Z1

sn− Z2

sn‖]dsn (49)

≤ (3L)n T n

n!

∫ T

0E[‖Z1

s − Z2s‖]ds (50)

From this we get that, for sufficiently large values of n ∈ IN , the operator Sn isa contraction operator on LT

1 and has therefore a unique fixed point. Supposethat Sn0 is a contraction operator on LT

1 with fixed point (Zt(ω))t≥0. We get

∫ T

0dtE[‖Zt − SZt‖] =

∫ T

0dtE[‖Skn0Zt − Skn0+1Zt‖] (51)

≤ 3kn0Lkn0T kn0

kn0!

∫ T

0dtE[‖Zt − SZt‖] → 0 when k →∞ , (52)

so that (Zt(ω))t≥0 is a fixed point also for the operator S and solves equation(1).

Proof of Corollary 4.4: The statement in Corollary 4.4 is a direct consequenceof Theorem 4.3 and Proposition 3.15. From the proof of Theorem 4.3 it followsin fact that Mt(ω) :=

∫ t

0

∫Λf(s, x, Zs(ω), ω)q(dsdx)(ω) is a strong -1- integral.

Proof of Theorem 4.6: Let (Zt(ω))t∈[0,T ]∈ LT2 . Similar to the proof of Theo-

rem 4.3 (by taking p = 2 instead of p = 1), it can be proven that A(s, Zs(ω), ω)is Bochner integrable w.r.t. the Lesbegues measure ds on [0, T ], for P -a.e.ω ∈ Ω. In fact,∫ T

0

‖A(s, Zs(ω), ω)‖2ds ≤ K

(T +

∫ T

0

‖Zs(ω)‖2ds

). (53)

That f(s, x, Zs(ω), ω) is strong -2-integrable w.r.t. q(dsdx)(ω) on (0, T ]×Λ fol-lows from (39), Theorem 3.12, resp. Theorem 3.13, and the following inequality∫ T

0

∫Λ

E[‖f(s, x, Zs(ω), ω)‖2]ν(dsdx) ≤ K

(T +

∫ T

0

E[‖Zs(ω)‖2]ds

). (54)

Let S denote the operator defined in (40). Similar to the proof of Theorem 4.3,it can be proven (by taking LT

2 instead of LT1 ), that (Zt(ω))t∈[0,T ]∈ LT

2 implies(SZt(ω))t∈[0,T ] ∈ LT

2 .

We shall prove that S is a contraction operator on LT2 .

Let Z1t (ω) ∈ LT

2 and Z2t (ω) ∈ LT

2 .

Let K2 = 1 if F is a separable Hilbert space and otherwise be the constant inthe definition of M -type 2 spaces. It follows from (38) and Theorem 3.12, resp.Theorem 3.13, that

17

E[ ‖SZ1t (ω)− SZ2

t (ω)‖2 ]

≤ 2T∫ t

0E[‖A(s, Z1

s (ω), ω)−A(s, Z2s (ω), ω)‖2] ds

+2K2

∫ t

0

∫ΛE[‖f(s, x, Z1

s )− f(s, x, Z2s )‖2] ν(dsdx) ≤ (2 + 2K2)L

∫ t

0E[‖Z1

s − Z2s‖2]ds

It follows by induction that

∫ T

0E[ ‖SnZ1

t (ω)− SnZ2t (ω)‖2 ]dt (55)

≤ (2 + 2K2)nLn∫ T

0dt∫ t

0ds1

∫ s1

0ds2....E[‖Z1

sn− Z2

sn‖2]dsn (56)

≤ (2 + 2K2)nLn T n

n!

∫ T

0E[‖Z1

s − Z2s‖2]ds (57)

The rest of the proof is similar to the proof of Theorem 4.3.

Proof of Corollary 4.7: Corollary 4.7 is a straight consequence of Theorem4.6 and Remark 3.9.

Remark 4.8 Suppose that the conditions in Theorem 4.3, resp. 4.6 are satis-fied. Using the inequalities (37), resp. (39), and Gronwall’s Lemma it followsthat there are constants CT,K and LT,K depending on T and K such that thesolution (Zt(ω))t∈[0,T ] of (59) satisfies∫ t

0

E[‖Zs(ω)‖p ds ≤ LT,K

∫ t

0

E[‖Φ(s, ω)‖p] ds + CT,K (58)

In the above theorems we have produced a solution which is Ft -adapted andis in LT

p , which has cad -lag version. Howevever in case Φ(t, ω)= Φ(ω) we canproduce a solution which is in D([0, T ], E) and the uniqueness in this case is inthe sense of P (Z1(t, ω) = Z2(t, ω),∀t ∈ [0, T ])= 1. Let us consider the stochasticdifferential equation

Zt(ω) = Φ(ω)+∫ t

0

A(s, Zs(ω), ω)ds+∫ t

0

∫Λ

f(s, x, Zs(ω), ω)(N(dsdx)(ω)−ν(dsdx))

(59)

Theorem 4.9 Fix p = 1 or p = 2. If p = 2 assume that F is a separableBanach space of M -type 2. Let 0 < T <∞. Suppose that

E[‖φ(ω)‖p] <∞

Suppose that there exists a constant L > 0 and a constant K > 0, s.th. (36)and (37) (if p = 1), resp. (38) and (39) (if p = 2) holds. Then there exists aunique process satisfying (59) s.th.

18

a) Zt is cad -lag,b) Zt is Ft -measurable,c)

∫ T

0E[‖Zs‖p] <∞

Proof of Theorem 4.9:Proof of the Uniqueness:Let p = 1. Let (Zt)t∈[0,T ] and (Zt)t∈[0,T ] be two solutions with initial conditionφ(ω), resp. φ(ω) satisfying a), b) and c) above. Then

E[‖Zt(ω)− Zt(ω)‖] ≤ E[‖Φ(ω)− Φ(ω)‖]+∫ t

0E[‖A(s, Zs(ω), ω)−A(s, Zs(ω), ω)‖]ds

+2∫ t

0

∫ΛE[‖f(s, x, Zs(ω), ω)− f(s, x, Zs(ω), ω)‖]β(dx)ds

≤ E[‖Φ(ω)− Φ(ω)‖] + 2L∫ t

0E[‖Zs(ω)− Zs(ω)‖]ds , (60)

where we used inequality (36) and Theorem 3.10. Let

v(s) := E[‖Zs(ω)− Zs(ω)‖]

then

v(t) ≤ E[‖Φ(ω)− Φ(ω)‖]e2Lt

We get that if Φ(ω) = Φ(ω) then

P (Zt(ω) = Zt(ω), t ∈ Q ∩ [0, T ]) = 1

where with Q we denote the rational numbers. By the cad -lag property weget uniqueness. We get also continuity of the solution of (59) w.r.t. the initialcondition. The proof works in a similar way for p = 2 .

Proof of the Existence:Let

Z0t (ω) = Φ(ω) ∀t ∈ [0, T ] (61)

and

Zk+1t (ω) = Φ(ω) +

∫ t

0A(s, Zk

s (ω), ω) ds

+∫ t

0

∫Λf(s, Zk

s (ω), x, ω)q(dsdx)(ω) (62)

From (26) and (36) we get that for any k ∈ IN , t ∈ [0, T ],

E[‖Zk+1t (ω)− Zk

t (ω)‖] ≤ 2L∫ t

0

E[‖Zks (ω)− Zk−1

s ‖]ds (63)

19

and

E[‖Z1t (ω)− Z0

t (ω)‖] ≤∫ t

0E[‖A(s, Z0

s (ω), ω)‖] ds

+E[‖∫ t

0

∫Λf(s, Z0

s (ω), x, ω)q(dsdx)(ω)‖]

≤ 2K∫ t

0(1 + E[‖Φ(ω)‖])ds ≤ 2Kt(1 + E[‖Φ(ω)‖]) (64)

Repeating in k, we get

E[‖Zk+1t (ω)− Zk

t (ω)‖] ≤ (2t)k+1

(k + 1)!LkK(1 + E[‖Φ(ω)‖]) (65)

We have

sup0≤t≤T ‖Zk+1t (ω)− Zk

t (ω)‖ ≤∫ T

0‖A(s, Zk+1

s (ω), ω)−A(s, Zks (ω), ω)‖ds

+sup0≤t≤T ‖∫ t

0

∫Λ

(f(s, Zks (ω), x, ω)− f(s, Zk−1

s (ω), x, ω)) q(dsdx)(ω)‖(66)

It follows

P (sup0≤t≤T ‖Zk+1t (ω)− Zk

t (ω)‖ > 2−k)

≤ P (∫ T

0‖A(s, Zk

s (ω), ω)−A(s, Zk−1s (ω), ω)‖ds > 2−k−1)

+P (sup0≤t≤T ‖∫ t

0

∫Λ

(f(s, Zks (ω), x, ω)− f(s, Zk−1

s (ω), x, ω)) q(dsdx)(ω)‖ > 2−k−1)

≤ 2k+1E[∫ T

0‖A(s, Zk

s (ω), ω)−A(s, Zk−1s (ω), ω)‖ds]

+2k+1E[‖∫ T

0

∫Λ

(f(s, Zks (ω), x, ω)− f(s, Zk−1

s (ω), x, ω)) q(dsdx)(ω)‖]

≤ 2k+1E[∫ T

0‖A(s, Zk

s (ω), ω)−A(s, Zk−1s (ω), ω)‖ds]

+2k+1+1∫ T

0

∫ΛE[‖f(s, Zk

s (ω), x, ω)− f(s, Zk−1s (ω), x, ω)‖]dsβ(dx)

≤ 2k+1 (2t)k+1

(k+1)! LkK(1 + E[‖Φ(ω)‖]) (67)

where the last inequality is obtained in a similar way as (65). By Borel -CantelliLemma we get that P -a.s. there exists k0(ω) ∈ IN , s. th.

sup0≤t≤T

‖Zk+1t (ω)− Zk

t (ω)‖ ≤ 2−k ∀k ≥ k0(ω) (68)

Define

Znt (ω) = Z0

t (ω) +n−1∑k=0

(Zk+1t (ω)− Zk

t (ω)) (69)

Then Znt converges P - a.s. uniformly on [0, T ].

Let

Zt(ω) := limn→∞

Znt (ω) (70)

20

then Zt(ω)t∈[0,T ] is cad -lag, as each Znt (ω)t∈[0,T ] is, and the limit is

in sup norm. Zt is Ft -measurable as Z1t , Z2

t ,..., Znt ,...are Ft -measurable by

induction.Note that for n > m

E[‖Znt − Zm

t ‖] ≤ E[‖∑n−1

m (Zk+1t − Zk

t )‖]

≤∑n−1

m E[‖Zk+1t − Zk

t ‖] ≤ K(1 + E[‖Φ(ω)‖])∑∞

m(4t)k+1

(k+1)! Lk

→ 0 as m→∞ (71)

It follows that convergence holds also in L1, i.e.1

limn→∞

Znt = Zt

so that (Zt)t∈[0,T ] satisfies also c). The proof works in a similar way for p = 2.

Let us prove that (Zt)t∈[0,T ] satisfies (59).

Zn+1t (ω) = Φ(ω)+

∫ t

0

A(s, Zns (ω), ω)ds+

∫ t

0

∫Λ

f(s, x, Zns (ω), ω)(N(dsdx)(ω)−ν(dsdx))

(72)As (Zn

t )t∈[0,T ] converges uniformly on [0, T ] P -a.e. to (Zt)t∈[0,T ] it follows fromFatou’s Lemma that

E[∫ T

0

‖Zt − Znt ‖dt] ≤ lim

m→∞E[∫ T

0

‖Zmt − Zn

t ‖dt] = 0 (73)

We get from (28) and the Lipshitz property

E[‖∫ t

0

∫Λ

F (s, x, Zns (ω), ω)q(dsdx)(ω)−

∫ t

0

F (s, x, Zs(ω), ω)q(dsdx)(ω)‖ ≤ 2LE[∫ t

0

∫Λ

‖Zns (ω)−Zs(ω)‖ds

(74)so that

1

limn→∞

∫ t

0

∫Λ

F (s, x, Zns (ω), ω)q(dsdx)(ω) =

∫ t

0

∫Λ

F (s, x, Zs(ω), ω)q(dsdx)(ω)

(75)Similarly

1

limn→∞

∫ t

0

A(s, Zns (ω), ω)ds =

∫ t

0

A(s, Zs(ω), ω)ds (76)

There exists a subsequence such that the convergence in (75) and (76) holds P-a.s.. Thus (Zt)t∈[0,T ] solves P -a.e. (59). The proof works in a similar way forp = 2.

Theorem 4.10 Fix p = 1 or p = 2. If p = 2 assume that F is a separableBanach space of M -type 2. Let 0 < T < ∞. Suppose that for every constantC > 0 there is a constant LC > 0, s.th.

21

T‖A(t, z, ω)−A(t, z′, ω)‖p +∫Λ‖f(t, x, z, ω)− f(t, x, z′, ω)‖pβ(dx)

≤ LC‖z − z′‖p for all t ∈ (0, T ] ,for all z, z′ ∈ F s.th. ‖z‖ ≤ C, ‖z′‖ ≤ C ,

and P − a.e. ω ∈ Ω . (77)

Assume also that there is a constant K such that

‖A(t, z, ω)‖p +∫Λ‖f(t, x, z, ω)‖pβ(dx) ≤ K(‖z‖p + 1) for all t ∈ (0, T ] ,for all z ∈ F , and P − a.e. ω ∈ Ω . (78)

LetE[ sup

t∈[0,T ]

‖φ(t, ω)‖p] <∞ . (79)

Then there is, up to stochastic equivalence, a unique Ft -adapted process (Zt)0≤t≤T

∈ LTp , which satisfies (1). Assume also that (φ(t, ω))0≤t≤T has P -a.s. no dis-

continuities of the second kind, then (Zt)0≤t≤T has no discontinuities of thesecond kind. If (φ(t, ω))0≤t≤T is cad -lag, then (Zt)0≤t≤T is cad -lag.Suppose that φ(t, ω) = φ(ω), i.e. φ does not depend on time. Then there existsa unique process which satisfies (1), s.th. a), b), and c) in Theorem 4.9 holds.

To prove this theorem we follow the strategy proposed in [27] to generalize theproof of the existence of a unique solution of a Hilbert valued SDE driven bya Gaussian noise with drift terms satisfying Lipshitz conditions, to the case oflocal Lipshitz conditions.

Proof: We denote with Bn := B(0, n) a centered ball in F with radius n , andwith d(z,Bn) the distance of z ∈ F from Bn . We also denote with

An(s, z, ω) := A(s,z

1 + d(z,Bn), ω) (80)

fn(s, x, z, ω) := f(s, x,z

1 + d(z,Bn), ω) (81)

An and fn satisfy conditions A), B), the Lipshitz condition (36) (resp. (38)),and the growth condition (37) (resp. (39)), if p = 1 (resp. p = 2).

It follows from Corollary 4.4, resp. 4.7, that for each n ∈ IN there is, up tostochastic equivalence, a unique Ft -adapted process (Zn

t )0≤t≤T ∈ LTp , which

satisfies

Znt (ω) = φ(t, ω)+

∫ t

0

An(s, Zns (ω), ω)ds+

∫ t

0

∫Λ

fn(s, x, Zns (ω), ω)(N(dsdx)(ω)−ν(dsdx))

(82)

22

Moreover, (Znt (ω))0≤t≤T has no discontinuities of the second kind, resp. is cad

-lag, if this holds for (φ(t, ω))0≤t≤T . If φ(t, ω) = φ(ω) then for each n ∈ INthere exists a unique process (Zn

t (ω))0≤t≤T which satisfies (1), s.th. a), b), andc) in Theorem 4.9 holds.

Let Tn := supt : ‖Zn+1t (ω))‖ ≤ n, then Tn is an Ft -stopping time and

Zn+1t∧Tn

(ω) = φ(t ∧ Tn, ω) +∫ t∧Tn

0An+1(s, Zn+1

s∧Tn(ω), ω)ds

+∫ t∧Tn

0

∫Λfn+1(s, x, Zn+1

s∧Tn(ω), ω)(N(dsdx)(ω)− ν(dsdx))

= φ(t ∧ Tn, ω) +∫ t∧Tn

0A(s, Zn+1

s∧Tn(ω), ω)ds

+∫ t∧Tn

0

∫Λf(s, x, Zn+1

s∧Tn(ω), ω)(N(dsdx)(ω)− ν(dsdx))

= φ(t ∧ Tn, ω) +∫ t∧Tn

0An(s, Zn+1

s∧Tn(ω), ω)ds

+∫ t∧Tn

0

∫Λfn(s, x, Zn+1

s∧Tn(ω), ω)(N(dsdx)(ω)− ν(dsdx)) . (83)

It follows that Zn+1t∧Tn

= Znt∧Tn

P -a.s., ∀s ∈ [0, T ], which implies that Tn ≥Tn−1 , P -a.s.. Moreover, we shall prove that

P (∀n , Tn < t) = 0 (84)

It then follows that there is a process (Zt(ω))t∈[0,T ], which is Ft -adapted, is inLT

p , and s.th.

Zt∧Tn= Zn+1

t∧TnP − a.s. (85)

limn→∞

Zn+1Tn∧t = Zt(ω) P − a.s. ∀t ∈ [0, T ] (86)

and

Zt∧Tn(ω) = φ(t ∧ Tn, ω) +∫ t∧Tn

0A(s, Zs∧Tn(ω), ω)ds

+∫ t∧Tn

0

∫Λf(s, x, Zs∧Tn

(ω), ω)(N(dsdx)(ω)− ν(dsdx)) . (87)

As a consequence (Zt)t∈[0,T ] is, up to stochastic equivalence, the unique Ft

-adapted process in LTp , which satisfies (1). Moreover, (Zt(ω))0≤t≤T has no dis-

continuities of the second kind, resp. is cad -lag, if this holds for (φ(t, ω))0≤t≤T .Moreover if φ(t, ω) = φ(ω), then (Zt)t∈[0,T ] is the unique process which satisfies(1), s.th. a), b), and c) in Theorem 4.9 holds.

We now prove (84):

23

P (∀n , Tn < t) ≤ P (Tn < t) ≤ P (sups≤t ‖Zn+1s ‖ > n)

≤ P (sups≤t ‖φ(s, ω)‖ > n/3) + P (sups≤t ‖∫ s

0An+1(s′, Zn+1

s′ (ω), ω)ds′‖ > n/3)

+P (sups≤t ‖∫ s

0

∫Λfn+1(s′, x, Zn+1

s′ (ω), ω)q(ds′dx)(ω)‖ > n/3)

≤ 3p E[sups≤t ‖φ(s,ω)‖p]

np + 3p E[sups≤t ‖∫ s0 An+1(s

′,Zn+1s′ (ω),ω)ds′‖p]

np

+3p sups≤tE[‖

∫ s0

∫Λ fn+1(s

′,x,Zn+1s′ (ω),ω)q(dsdx)(ω)‖p]

np

≤ 3p E[sups≤t ‖φ(s,ω)‖p]

np +6p2K(t+

∫ t0 E[‖Zn+1

s′ ‖pds′])

np → 0 when n→∞ (88)

where in the fourth inequality we used Doob’s inequality for the martingale∫ s

0

∫Λfn+1(s′, x, Zn+1

s′ (ω), ω)q(ds′dx)(ω), in the last inequality we used the growthcondition (37), resp. (39), and Theorems 3.10- 3.13.

5 Continuous dependence on initial data andMarkov property

In this Section we analyze the continuous dependence of the solutions of (1) fromthe initial condition, as well as from the drift and noise coefficient (Theorem5.1 below). We then analyze the Markov property (Theorem 5.2 below). Weassume again that (E,B(E)), and (F,B(F )) are separable Banach spaces anduse the same notations as in the previous Sections. We continue assuming inthe whole article that the compensator ν(dtdx) of the Poisson random measureN(dtdx)(ω) is a product measure on B(IR+ × E), i.e. ν(dtdx) = α(dt)β(dx),such that hypothesis A in Theorem 3.5 is satisfied, and that α(dt) = dt. Wealso assume that f0(t, x, z, ω) := f(t, x, z, ω), A0(t, z, ω) := A(t, z, ω), φ0(t, ω):= φ(t, ω) satisfy the conditions A), resp. B) resp. C) in Section 4. Moreover,we assume that this holds also for fn(t, x, z, ω), resp. An(t, z, ω), resp. φn(t, ω),where n ∈ IN . We prove the following result

Theorem 5.1 Fix p = 1 or p = 2. If p = 2 assume that F is a Banach spaceof M -type 2. Let T > 0. Assume that there is a constant K > 0 such that forall n ∈ IN0, t ∈ [0, T ] and z ∈ F

‖An(t, z, ω)‖p +∫

Λ

‖fn(t, x, z, ω)‖pβ(dx) ≤ K(‖z‖p + 1) . (89)

Assume that for any C > 0 there is a constant LC such that for all n ∈ IN0,t ∈ [0, T ] and z , z′ ∈ F , with ‖z‖ < C, ‖z′‖ < C,

T‖An(t, z, ω)−An(t, z′, ω)‖p+∫

Λ

‖fn(t, x, z, ω)−fn(t, x, z′, ω)‖pβ(dx) ≤ LC‖z−z′‖p .

(90)

24

Moreover, assume that

supn∈IN0

E[ supt∈[0,T ]

‖φn(t, ω)‖p] <∞ . (91)

and that for all t ∈ [0, T ] and z ∈ F

‖φn(t, ω)− φ(t, ω)‖+ ‖An(t, z, ω)−A(t, z, ω)‖+∫Λ‖fn(t, x, z, ω)− f(t, x, z, ω)‖β(dx) → 0in probability as n→∞ . (92)

Let us denote with Zn(t, ω) the solutions of (1) for the case where the initialcondition, resp. the drift, resp. the noise coefficient are φn(t, ω), An(t, z, ω),fn(t, x, z, ω).

Then for each t ∈ [0, T ], Zn(t, ω) converge in probability to Z(t, ω) when n→∞.

Proof of Theorem 5.1: We first remark that from the assumptions (89),(90),(91), and Theorem 4.10 it follows that (Zn(t))t∈[0,T ] exists and is uniquely de-fined up to stochastic equivalence. Moreover from the assumptions (89), (91),it follows that

E[ supt∈[0,T ]

‖Zn(t)‖] ≤ eCTE[ supt∈[0,T ]

‖φn(t)‖] (93)

Let us define

ψNn (t, ω) = 1 if ‖φn(s, ω)‖+ ‖φ(s, ω)‖+ ‖Zn(s, ω)‖+ ‖Z(s, ω)‖ ≤ N ∀s ∈ [0, t]

= 0 if ‖φn(s, ω)‖+ ‖φ(s, ω)‖+ ‖Zn(s, ω)‖+ ‖Z(s, ω)‖ > N ∀s ∈ [0, t]

Let

(Zn(t, ω)− Z(t, ω))ψNn (t, ω) = (φn(t, ω)− φ(t, ω))ψN

n (t, ω)

+ψNn (t, ω)

∫ t

0(An(s, Zs(ω), ω)−A(s, Zs(ω), ω)ds

+∫ t

0

∫Λ(fn(s, x, Zs(ω), ω)− f(s, x, Zs(ω), ω))q(dsdx)(ω) (94)

Since ψNn (t, ω) ≤ ψN

n (s, ω) ∀s, t ∈ [0, T ], s ≤ t, it follows from the assumption(89) that there is a constant C > 0, s.th.

E[‖Zn(t, ω)− Z(t, ω)‖pψNn (t, ω)] ≤ E[αN

n (t, ω)]

+C∫ t

0E[‖Zn(s, ω)− Z(s, ω)‖ψN

n (s, ω)] ds (95)

25

where

αNn (t) := ‖φn(t, ω)− φ(t, ω)‖pψN

n (t, ω) (96)

We remark that from (92) it follows that αNn (t, ω) → 0 in probability, as n→

∞ , uniformly in t ∈ [0, T ]. From the Lesbegues dominated convergence theoremand

‖φn(t, ω)− φ(t, ω)‖pψNn (t) ≤ 2pNp . (97)

it follows that

E[αNn (t, ω)] → 0 as n→∞ uniformly in t ∈ [0, T ] (98)

By Gronwall’s Lemma (see e.g. [11]) it follows that

E[‖Zn(t, ω)− Z(t, ω)‖pψNn (t)] → 0 as n→∞ . (99)

Let ε > 0, then

P (‖Zn(t, ω)−Z(t, ω)‖ > ε) ≤ 1εE[ψN

n (t, ω)‖Zn(t, ω)−Z(t, ω)‖]+P (ψNn (t, ω) = 0)

(100)

P (ψNn (t, ω) = 0) ≤ P (sups∈[0,t]‖φn(s, ω)‖ > N/4) + P (sups∈[0,t]‖φ(s, ω)‖ > N/4)

+P (sups∈[0,t]‖Zn(s, ω)‖ > N/4) + P (sups∈[0,t]‖Z(s, ω)‖ > N/4) , (101)

so that from (91), (93) it follows that

P (ψNn (t, ω) = 0) → 0 as N → 0 uniformly in n ∈ IN (102)

and hence

limN→∞ lim supn→∞ P (‖Zn(t, ω)− Z(t, ω)‖ > ε)≤ 1

εp limN→∞ lim supn→∞E[ψNn (t, ω)‖Zn(t, ω)− Z(t, ω)‖p]

+ limN→∞ lim supn→∞ P (ψNn (t, ω) = 0) , (103)

the left hand side equal zero, completing the proof.

Theorem 5.2 Let A(t, z, ω) = A(z) and φ(t, ω) =φ(ω), for all t ∈ [0,∞) .Fix p = 1 or p = 2. If p = 2 assume that F is a Banach space of M -type 2.Let T > 0. Assume also that the hypothesis in Theorem 4.10 are satisfied andlet (Zφ

t )t∈IR be the solution of (1). Let (Zzt )t∈IR , z ∈ F , be the solution of (1),

if φ(ω) = z ∀ω ∈ Ω.Thena) (Zφ

t )t≥0 is time homogenousb) (Zz

t )t≥0 is Markov.

26

Proof of Theorem 5.2: Let us denote with (Zs,φt )t≥s the solution of

Zs,φt (ω) = φ(ω) +

∫ t

s

A(Zs,φu (ω))du+

∫ t

s

∫Λ

f(x,Zs,φu (ω))q(dudx)(ω) (104)

Following the proof of Theorem 4.10 it can be checked that such solution existsand is unique up to stochastic equivalence. Let us remark that the compensatedLevy random measure q(dsdx)(ω) is translation invariant in time. I.e. if h > 0and q(dsdx)(ω) denotes the unique σ -finite measure on B(IR+×E \0) whichextends the pre -measure q(dsdx)(ω) on S(IR+)×B(E\0), such that q((s, t],Λ):= q((s+ h, t+ h],Λ), for (s, t]× Λ ∈ S(IR+)×B(E \ 0), then q(A) and q(A)are equally distributed for all A ∈ B(IR+ × E \ 0).It follows that

Zs,φs+h(ω) = φ(ω) +

∫ s+h

sA(Zs,φ

u (ω))du+∫ s+h

s

∫Λf(x,Zs,φ

u (ω))q(dudx)(ω)

= φ(ω) +∫ h

0A(Zu,φ

s+u(ω))du+∫ h

0

∫Λf(x, Zs,φ

s+u(ω))q(dudx)(ω) (105)

From Theorem 4.10 it follows

Z0,φh (ω) = φ(ω) +

∫ h

0

A(Z0,φu (ω))du+

∫ h

0

∫Λ

f(x,Z0,φu (ω))q(dudx)(ω) (106)

As the solutions of (105) and (106) are unique up to stochastic equivalent andq(dsdx) and q(dsdx) are equal distributed, it follows that (Z0,φ

h (ω))h≥0 and(Zs,φ

s+h(ω))h≥0 are stochastic equivalent. This proves property a).

We remark that from Theorem 4.10 it follows that (Z0,φt (ω))t≥0 is cad -lag.

Let T ≥ 0. We denote by Qφ the distribution induced by (Z0,φt (ω))t∈[0,T ] on

the Skorohod space D([0, T ], F ), and by Eφ the corresponding expectation. Wealso remark that the σ -algebra Fφ

t := σZ0,φs , s ≤ t ⊆ Ft, where (Ft)t≥0,

denotes the natural filtration of the compensated Levy measure q(dsdx)(ω),and σZ0,φ

s , s ≤ t is the σ -algebra generated by (Z0,φs )s≤t.

Let us consider now the solution (Zr(ω))r∈[t,T ] of

Zr(ω) = Zt(ω) +∫ r

t

A(Zu(ω))du+∫ u

t

∫A

f(x,Zu(ω))q(dudx)(ω) (107)

From Theorem 4.10 it follows that (Zr(ω))r∈[t,T ] is stochastic equivalent to(Zt,Zt

r (ω))r∈[t,T ]. Let us define H(z, t, r, ω):= Zt,zr (ω), r ∈ [t, T ]. We remark

that H(z, t, r, ω) is independent of Ft.

Consider γ bounded, real valued measurable function on F . Then we can write

Ez[γ(Zt+h)|Ft](ω) = E[γ(H(Zt, t, t+ h, ω))|Ft](ω) (108)

27

and

EZt(ω)[γ(Zh(ω))] = E[γ(H(z, 0, h, ω))]z=Zt(ω) , (109)

where E[·]z=Zt(ω) := E[·|Zt(ω) = z]. We shall prove that

E[γ(H(Zt, t, t+ h, ω))|Ft](ω) = E[γ(H(z, t, t+ h, ω))]z=Zt(ω) (110)

It then follows by property a) and (109) that

E[γ(H(Zt(ω), t, t+ h, ω))|Ft](ω) = E[γ(H(z, t, t+ h, ω))]z=Zt(ω)

= E[γ(H(z, 0, h, ω))]z=Zt(ω) = EZt(ω)[γ(Zh(ω))] (111)

and hence, using (108), it follows

Ez[γ(Zt+h(ω)|Ft] = EZt(ω)[γ(Zh(ω))] (112)

and, since FZtt ⊆ Ft this gives

Ez[γ(Zt+h(ω)|FZtt ] = EZt(ω)[γ(Zh(ω))] (113)

Proof of (110):Put g(z, ω) = γ(H(z, t, t + h, ω)) . Clearly g(z, ·) is measurable in ω , andz → g(z, ω) is continuous by the continuity with respect to the initial condition.Thus g(z, ω) is separately measurable, since F is separable. It follows thatg(z, ω) is jointly measurable. Clearly g is bounded. We can approximate gpointwise boundedly by functions of the form

∑mk=r φk(z)ψk(ω).

E[g(Zt(ω), ω)|Ft] = limm→∞∑m

k=1 φk(Zt(ω))E[ψk(ω)|Ft]= limm→∞

∑mk=1E[φk(z)ψk(ω)|Ft]z=Zt(ω) = E[g(z, ω)]z=Zt(ω) (114)

where in the first inequality we used that φk(Zt(ω)) is Ft -measurable.

6 APPENDIX: Stochastic integrals w.r.t. Levyprocesses

We assume again that (E,B(E)), and (F,B(F )) are separable Banach spacesand use the same notations as in the previous Sections. Let p = 1, or p = 2 and(F,B(F )) be a separable Banach space of type 2. We assume that∫

E\0‖f(x)‖pβ(dx) <∞ (115)

28

then

ξt :=∫ t

0

∫‖f(x)‖≤1

f(x)q(dsdx) +∫ t

0

∫‖f(x)‖>1

f(x)N(dsdx)(ω)

=∫ t

0

∫E\0 f(x)q(dsdx) +

∫ t

0

∫‖f(x)‖>1

f(x)β(dx)ds (116)

is a Markov process (Theorem 5.2). In this Section we show that any suchsolution (ξt)t≥0 is a Levy process. This implies in particular, by taking f(x) = x,that any compensated Poisson - Levy measure is associated to a Levy process.

Let us apply the Ito formula [22]

H(ξt(ω))−H(ξτ (ω)) =∫ t

τ

∫0<‖f(x)‖≤1

H(ξs−(ω) + f(x))−H(ξs−(ω)) q(dsdx)(ω)

+∫ t

τ

∫0<‖f(x)‖≤1

H(ξs−(ω) + f(x))−H(ξs−(ω))−H′(ξs−(ω))f(x) dsβ(dx)

+∫ t

τ

∫‖f(x)‖>1 H(ξs−(ω) + f(x))−H(ξs−(ω))N(dsdx)(ω)

P − a.s. , (117)

to H(ξt(ω)) = eix?(ξt), for x? ∈ E?, E? denoting the dual space of E. (In [22]the Ito formula is given in a much more general statement) We get

E[eix?(ξ(t)) − 1] =∫ t

0

∫E\0 E[eix?(ξ(s))

(eix?(f(x)) − 1− ix?(f(x))1‖f(x)‖≤1

)dsβ(dx) (118)

Define φt(x?) = E[eix?(ξ(t))], then we get

d

dtφt(x?) = cφt(x?) (119)

φ0(ξ?) = 1 (120)

with

c :=∫

E\0

(eix?(f(x)) − 1− ix?(f(x))1‖f(x)‖≤1

)β(dx) (121)

Solving the above equation, we get

φt(x?) = et∫

E\0

(eix?(f(x))−1−ix?(f(x))1‖f(x)‖≤1β(dx)

)(122)

Thus (ξt)t≥0, solution of (116) is a Levy process with Levy measure βf (dx), suchthat βf (A) =

∫Aβ(f−1(dx)) for any A ∈ B(E \0). Conversely, from the Levy

-Ito decomposition Theorem [2] it follows that, given any Levy process (ξt)t≥0,

29

such that the associated Levy measure β(dx) satisfies (115) for f(x)= x , then(ξt)t≥0 satisfies (116), with f(x) = x. (If however we interpret the stochasticintegral w.r.t the compensated Poisson measure q(dsdx) to be a simple -p -integral, in the sense of [21], for any p ≥ 1, then the Levy -Ito decomposition(116), with f(x) = x holds for any Levy process having values on any separableBanach space, the condition (115) not being necessary [8]).

7 APPENDIX: Examples of applied problemscoming from finance

We now prove that the equation (124) below, which comes as a model for volatil-ity in Finance [5], can be studied on general Banach spaces. Also for the casewhere the state space is the real line, some of the results can be obtained in amore direct way than [23] and [14] (see Remark 7.4 below).

Let p = 1, or p = 2 and (F,B(F )) be a separable Banach space of type 2. Weassume that ∫

0<‖x‖≤1

‖x‖pβ(dx) <∞ (123)

We analyze

dηt(ω) = −aηt(ω)dt+ dξt(ω) (124)

where

ξt(ω) :=∫ t

0

∫0<‖x‖≤1

xq(dsdx)(ω) +∫ t

0

∫‖x‖>1

xN(dsdx)(ω) , (125)

a > 0, ν(dsdx) = dsβ(dx) and with the initial condition η0 being independentof the filtration (Ft)t≥0 of (ξt)t∈[0,T ], and where we define

dξt(ω) =∫

0<‖x‖≤1

xq(dsdx)(ω) +∫‖x‖>1

xN(dsdx)(ω) (126)

From the results in the previous Sections (Theorem 4.10, Theorem 5.2) we knowthat for every T > 0 there is a unique path wise solution (ξt)t∈[0,T ] of (124) withinitial condition η0. Moreover if η0 = x, x ∈ F , then (ξt)t∈[0,T ] is Markov. Itcan be shown, using Ito formula ([22]), that the solution is

ηt(ω) = e−at(η0(ω) +∫ t

0

easdξs(ω)) (127)

In fact, applying the Ito formula [22]

30

H(t, Yt(ω))−H(τ, Yτ (ω)) =

+∫ t

τ∂sH(s, Ys−(ω))ds+

∫ t

τ

∫0<‖x‖≤1

H(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω)) q(dsdx)(ω)

+∫ t

τ

∫0<‖x‖≤1

H(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω))− ∂yH(s, Ys−(ω))f(s, x, ω) dsβ(dx)

+∫ t

τ

∫‖x‖>1

H(s, Ys−(ω) + f(s, x, ω))−H(s, Ys−(ω))N(dsdx)(ω)

P − a.s. for any 0 < τ < t ≤ T, (128)

to

H(s, z) := e−asz (129)

Ys(ω) := η0(ω) +∫ t

0

easdξs(ω)) (130)

f(s, x, ω) := easx (131)

we obtain

H(t, Yt(ω))−H(τ, Yτ (ω)) =

−a∫ t

τe−asYs(ω)ds

+∫ t

τ

∫0<‖x‖≤1

e−as[Ys−(ω) + easx]− e−asYs−(ω)− xdsβ(dx)

+∫ t

τ

∫0<‖x‖≤1

e−as[Ys−(ω) + easx]− e−asYs−(ω)q(dsdx)(ω)

+∫ t

τ

∫‖x‖>1

e−as[Ys−(ω) + easx]− e−asYs−(ω)N(dsdx)(ω)

= −a∫ t

τe−asYs(ω)ds

+∫ t

τ

∫0<‖x‖≤1

xq(dsdx)(ω) +∫ t

τ

∫‖x‖>1

xN(dsdx)(ω) (132)

and hence (127).

dηt(ω) = −ae−at

(η0(ω) +

∫ t

0

easdξs(ω))dt+ e−atd

(∫ t

0

easdξs(ω))

(133)

where

d

(∫ t

0

easdξs(ω))

= easdξs(ω) (134)

Let us denote for x ∈ E \ 0, A ∈ B(E \ 0)

Pt(x,A) = P (ηt ∈ A/η0 = x) (135)

the transition probability of ηt. ThenPt(x,A) = δe−atx ? Qt(A) ∀A ∈ E \ 0 (136)

31

where with ? we denote the convolution, δ is the Dirac measure, and Qt(·)is the distribution of

∫ t

0e−(a(t−s)dξs. We remark that this coincides with the

distribution of∫ t

0e−asdξs, as ξs has stationary independent increments. If ηt

has invariant measure, i.e. there exists a measure µ on E s.th.∫E

Pt(x,A)µ(dx) = µ(A) (137)

then ∫E

δe−atx ? Qt(A)µ(dx) = µ(A) (138)

Theorem 7.1 Let η0 be independent of the filtration (Ft)t≥0 of (ξt)t≥0, and

ηt = e−at

(η0 +

∫ t

0

easdξs

). (139)

Let (Pt)t≥0 denote the Markov semigroup associated to (ηt)t≥0. Suppose that µis a corresponding invariant measure, and let L(X) denote the law of a randomvariable X, then if

∫ t

0e−asdξs converges in law, then

limt→∞

L(∫ t

0

e−asdξs) = µ (140)

Proof of Theorem 7.1:

µ = Pt ? µ = µ(e−at·) ? L(∫ t

0

e−a(t−s)dξs) (141)

We note that

L(∫ t

0

e−asdξ(s)) = L(∫ t

0

e−a(t−s)dξs) (142)

Since a > 0, e−at → 0 when t → ∞, gives µ(e−at·) → δ0 when t → ∞, so thatthere is a measure ν (see e.g. [17]), s.th.

L(∫ t

0

e−asdξs) → ν when t→∞ (143)

and ν = µ.

Let us discuss when Theorem 7.1 can be used to find the invariant measureof the solution (127) of (124), (125). We first prove that Pt(x, ·) is infinitelydivisible

32

Lemma 7.2 ∫E\0 e

ix?(y)Pt(x, dy) = exp [e−atix?(x)]

exp[∫ t

0

∫0<‖y‖≤1

(eiea(t−s)x?(y) − 1− iea(t−s)x?(y))dsβ(dy)]

exp[∫ t

0

∫‖y‖>1

(eiea(t−s)x?(y) − 1)dsβ(dy)]

= exp[e−atix?(x) +

∫ t

0ψ(e−asx?)ds

](144)

where exp(tψ(x?)) is the Fourier transform of ξt.

Proof of Lemma: The proof is obtained by applying Ito formula to (127),similar to what is done in (122).

Remark 7.3 Lemma 7.2 implies that Pt(x, ·) is infinitely divisible and the cor-responding Levy measure βt(·) is such that

βt(A) =∫

E\0β(dy)

∫ t

0

1A(e−asy)ds , A ∈ B(IR+ × E \ 0) , (145)

while the corresponding shift is

γ := e−atx+∫

E\0β(dy)

∫ t

0

e−asy(10<‖y‖≤1e−asy − 10<‖y‖≤1(y)ds . (146)

(See e.g. [23] for the definitions and properties related to infinitely divisiblelaws on IRd, Chapt. 3, Pragraph 17 in particular for such computations, [17]on Banach spaces).

Remark 7.4 Lemma 7.2 was proven in Lemma 17.1 in Chapt.3, Paragr. 17,[23], for the real valued case, however using an approximation by simple func-tions. The existence of the solution of (124) has been proven in Paragr. 17 of[23], only for the case where (ξt)t∈[0,T ] is of bounded variation, (i.e. has bigjumps), while it has been proven on the real line in [14] in an ad hoc way, bydefining the stochastic integral of e−as w.r.t. dξs by assuming that an integrationby part formula holds, which follows now as a consequence of the Ito formula[22].

Following [23] Theorem 17.5, Chapt. 3, Pragraph 17, it can be shown that if∫‖y‖>2

log ‖y‖β(dy) <∞ (147)

then the limit distribution µ in (140) exists and its Fourier transform µ is

µ =∫ ∞

0

ψ(e−asx?)ds (148)

33

We note that µ is infinitely divisible as the limit of infinitely divisible Qt. Similarto [23] one can show that the corresponding Levy measure β and shift γ is

β(A) := limt→∞

1a

∫E

β(dy)∫ ∞

0

1A(e−sy)ds , (149)

γ :=1a

∫‖y‖>1

y

‖y‖β(dy) . (150)

8 APPENDIX: Examples coming from financeand insurance

We shall now consider another example [7] where the use of stochastic differentialequations with respect to non -Gaussian additive noise play a role in finance andinsurance. Let (ξt)t∈IR+ be a real valued Levy process with Levy measure β,like in (125). Using Ito formula [22] we get a SDE for eξt as follows.

(eξt − 1) =∫ t

0

∫0<‖x‖≤1

eξs− ex − 1q(dsdx)

+∫ t

0

∫0<‖x‖≤1

eξs− ex − 1− xdsβ(dx)

+∫ t

0

∫‖x‖>1

eξs− ex − 1N(dsdx) (151)

In fact (151) is obtained by applying (128) to H(s, ξs) =H(ξs) = eξs . We notethat with a(s, y) = ey

∫0<‖x‖≤1

ex − 1 − xβ(dx), f(s, y, x)= ey(ex − 1) thisequation is of the form studied in Skorohod ([24], page 45). Because a(s, y),f(s, y, x) satisfy Lipshitz conditions given there, we get eξt is a unique solutionof (151) and is Markov. Let (ηt)t∈IR+ be another pure jump Levy process. Thenusing Ito formula [22]

ζt = eξt

(ζ0 +

∫ t

0

e−ξsdηs

)(152)

is a solution of the equationdζt = deξt + dηt (153)

with ζ0 independent of ξt, ηt, t ≥ 0. Defining like in the previous Section dηt

through the Levy -Ito decomposition, we get

deξt = eξt−∫0<‖x‖≤1

(ex − 1)q(dtdx)

+eξt−∫‖x‖>1

(ex − 1)N(dtdx)

+eξt−∫0<‖x‖≤1

(ex − 1− x)β(dx) (154)

34

(deξt)(ζ0 +

∫ t

0eξs−dηs

)= eξt−

(ζ0 +

∫ t

0eξs−dηs

)×

[∫0<‖x‖≤1

(ex − 1)q(dtdx)

+∫‖x‖>1

(ex − 1)N(dtdx) +∫0<‖x‖≤1

(ex − 1− x)β(dx)]

= ζt− [∫0<‖x‖≤1

(ex − 1)q(dtdx)

+∫‖x‖>1

(ex − 1)N(dtdx) +∫0<‖x‖≤1

(ex − 1− x)β(dx)] (155)

giving

(dζt) = ζt−∫0<‖x‖≤1

(ex − 1)q(dtdx)

+ζt−∫‖x‖>1

(ex − 1)N(dtdx)

+ζt−∫0<‖x‖≤1

(ex − 1− x)β(dx)

+dηt (156)

The SDE (156) has a unique Markov solution with initial condition ζ0 = 0 .The corresponding transition functions are constant in x.

Hence (ζt)t≥0 is a Markov process and because of continuous dependence oninitial condition [24], we get that the transition semigroup is Feller. Furtherusing stationary independent increment property of ξt and ηt we get

L(∫ t

0

eξt−ξsdηs

)= L(Qt) , (157)

where

Qt :=∫ t

0

eξsdηs . (158)

Let us denote with Pt(x,A):= E[ζt ∈ A|ζ0 = x], A ∈ B(IR) and with PXt thedistribution of a process (Xt)t∈IR+ at time t, then

Pt(x,A) = Peξtx+∫ t0 eξ

sdηs(159)

If we assume that Peξtx → δ0, when t → ∞, then we get that if the invariantmeasure µ exists for ζt then µ = P∫∞

0 eξs dηs. In particular if ηs = s for all

s ∈ IR+, we get that µ= L(∫∞

0eξsds

)which is called perpetuity as the invariant

measure of the solution of

dζt = deξt + dt (160)ζo = x .

To compute the infinitesimal generator A of

35

ζt = eξt

(ξ0 +

∫ t

0

e−ξsds

). (161)

is useful, e.g. to compute the invariant measure analytically. To compute A weapply the Ito formula [22] to ζt. Let F a real valued function, F ∈ C2(IR).

F (ζt)− F (ζ0) =∫ t

0

∫0<‖x‖≤1

[F (ζs− + ζs−(ex − 1))− F (ζs−)]q(dsdx)

+∫ t

0

∫‖x‖>1

[F (ζs− + ζs−(ex − 1))− F (ζs−)]N(dsdx)

+∫ t

0

∫0<‖x‖≤1

[F (ζs− + ζs−(ex − 1))− F (ζs−)− F ′(ζs−)ζs−(ex − 1))]dsβ(dx)

+∫ t

0F ′(ζs−)[ζs−

∫0<‖x‖≤1

(ex − 1− x)β(dx) + 1]ds (162)

E[F (ζt)− F (ζ0)] =∫ t

0

∫0<‖x‖≤1

[F (ζs− + ζs−(ex − 1))− F (ζs−)− F ′(ζs−)ζs−(ex − 1)]β(dx)ds

+∫ t

0F ′(ζs−)

[ζs−

∫0<‖x‖≤1

(ex − 1− x)β(dx) + 1]ds (163)

It follows that the infinitesimal generator A acts on the functions F ∈ C2(IR)as

AF (y) =∫0<‖x‖≤1

[F (y + y(ex − 1))− F (y)− F ′(y)(ex − 1)y]β(dx)

+F ′(y)[y∫0<‖x‖≤1

(ex − 1− x)β(dx) + 1] (164)

AF (y) =∫0<‖x‖≤1

[F (exy)− F (y)− F ′(y)(ex − 1)y]β(dx)

+yF ′(y)∫0<‖x‖≤1

(ex − 1− x)β(dx) + F ′(y) (165)

Suppose that ∫0<‖x‖≤1

‖x‖β(dx) <∞ (166)

then

AF (y) =∫0<‖x‖≤1

[F (exy)− F (y)]β(dx)

−yF ′(y)∫0<‖x‖≤1

xβ(dx) + F ′(y) (167)

Acknowledgements We thank Sergio Albeverio for many useful discussionsrelated to this work. The second author thanks Marc Veraar for useful discus-sions related to this work in connection to [21] and Carlo Marinelli for usefulcomments. The support and hospitality of the ”Sonderforschungsbereich” SFB611 in Bonn, as well as of the University Koblenz -Landau, is gratefully ac-knowledged.

36

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39

Bestellungen nimmt entgegen:

Institut für Angewandte Mathematikder Universität BonnSonderforschungsbereich 611Wegelerstr. 6D - 53115 Bonn

Telefon: 0228/73 3411Telefax: 0228/73 7864E-mail: anke@iam.uni-bonn.de Homepage: http://www.iam.uni-bonn.de/sfb611/

Verzeichnis der erschienenen Preprints ab No. 160

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